So if I understand correctly, if I define the following:
N are the number of sampling points
T the time between two sampling points
NT the total time sampled
then
A signal with one cycle in the sampled time span NT has frequency \nu_{1}=\frac{1}{NT}
A signal with two cycles in the...
If I have a signal, sampled at N data points with a time-interval of T, does this restrict the frequency resolution I can obtain in Fourier space?
I understand that from the Nyquist-Shannon sampling theorem it follows that all information on the Fourier transform of a T-sampled signal is...
I've always struggled with the commonly used measures of the intensity of electromagnetic radiation and it's catching up to me lately. Suppose \bar{P}(R,\phi,\theta) is the Poynting vector of an electromagnetic field (in spherical coordinates) with norm...
I'm not sure what you mean by "unfixed co-ordinate system mathematics". You mean vector spaces without choosing a basis?
As for tensors, this is what I know: a tensor product of two vector spaces V and W both over field K is a pair (T,\otimes) where T a vector space over K and \otimes\colon...
I have an electromagnetic field with a Poynting vector that has the following form in spherical coordinates:
$$\bar{P}(R,\phi,\theta)=\frac{f(\phi,\theta)}{R^2}\bar{e}_{r}$$
The exact nature of f(\phi,\theta) is not known. Suppose I measure the flux of this vector field by a flat area...
Thanks for your suggestion. However, the coherency matrix definitely treats polarized, unpolarized and partially polarized radiation, just as the Mueller matrix does (and unlike the Jones matrix). For example the coherency matrix of unpolarized radiation is
J=\begin{bmatrix}
\frac{I}{2}&0\\...
The electric field of quasi-monochromatic, partially polarized light can be expressed by the following random process (Goodman, Statistical optics)
\bar{E}(t,\bar{x})=u_{x}(t,\bar{x})\bar{e}_{x}+u_{y}(t,\bar{y})\bar{e}_{y}
u_{x}(t,\bar{x})=\Psi_{x} e^{i(\bar{k}\cdot\bar{x}-\omega t)}...
I'm reading "Statistical Optics" by Goodman to understand how unpolarized light can be rigorously treated and there is more to it than just the sum of some monochromatic plane waves.
The links you gave me follow the usual reasoning. First they derive the result (i.e. the scattered intensity or the cross section) by assuming a monochromatic plane wave. This starts by solving the equation of motion of the electron as given in my original post (for a free electron \omega_{0}=0...
Elastic scattering from a bound electron is classically described by considering the driven, damped harmonic oscillator model for the motion of a bound electron in a classical em-wave. The (non-relativistic) equation of motion is written as...
The time averaged norm of the Poynting vector of this electromagnetic field (elliptically polarized light):
\begin{split}
\bar{E}(t,\bar{x})=&(\bar{E}_{0x}+\bar{E}_{0y}e^{i \delta})e^{\bar{k}\cdot\bar{x}-\omega t}\\
\bar{B}(t,\bar{x})=&\frac{1}{\omega}(\bar{k}\times\bar{E}(t,\bar{x}))...
Thermal diffuse scattering and the lowering of the Bragg peak intensities due to thermal motion (Debye Waller factor) is all derived from stating that the scattered intensity (which is the time-averaged norm of the Poynting vector) given by
I=c^{te}\left< K.K^{\ast}\right>_{t}
where...
To calculate the intensity of the scattered radiation from a crystal after irradiating with X-rays, one can add up all electromagnetic fields of the oscillating electrons (calculated using the Liénard–Wiechert potential). Taking the time-average of the norm of the Poynting vector of the...